Use division algorithm to show that the square of any positive integer is of the form 3p or 3p + 1.
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Use division algorithm to show that the square of any positive integer is of the form 3p or 3p + 1.
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Step-by-step explanation:
Let
′
a
′
be any positive integer and b=3,
Then, by division algorithm a=3q+r,
for some integer q⩾0 and 0⩽r⩽3.
So, a=3q,3q+1,3q+2.
Then, the square of positive integer,
a=3q
a
2
=(3q)
2
=9q
2
=3(3q
2
)
=3p
(Where p=3q
2
)
a=3q+1
a
2
=(3q+1)
2
=9q
2
+6q+1
=3(3q
2
+2q)+1
=3p+1
(Where p=3q
2
+2q)
a=3q+2
a
2
=(3q+2)
2
=9q
2
+12q+4
=3(3q
2
+4q+1)+1
=3p+1
(Where p=3q
2
+4q+1)
Since p is some positive integer.
∴ The square of any positive integer is of the form 3p or 3p+1.
Answer:
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